Mickey,
Good question. I assume you're talking about the web version. The web version uses your browser cache to store a local copy of your airframe. This allows it to already be there the next time you come to the site, even if you close your browser in the mean time. You can also save a copy to your hard drive at any time through "File -> Save As..." Those are the two ways your airframe can be stored on your local computer.
Whenever you select "Update" or "Save Results" in the analysis tab, your geometry is sent to our web server so that it can crunch some numbers. That geometry is stored locally in a text formatted JSON file. We do not currently have a method for automatically or systematically deleting those files, but we periodically clean them out to free up room on the server. Your designs are not shared with others. However, we do store them for a short time so that we can access them in the event that a user runs into an error or bug we need to track down.
Does this help?
And now you have me curious - what type of airframe are you developing? Would you be willing to post an image?